# Minimum value on an open continuous function

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In summary, the problem is asking to prove that a continuous function on (a,b) with limits of infinity at both endpoints must have a minimum on the interval. One approach to this is to consider different regions of the function and use the continuity of f to make a statement about the function in the "not close to a or b" region. Another helpful strategy is to try to construct a function that goes to infinity at a and b, but does not have a minimum on (a,b).

## Homework Statement

Suppose that f is a continuous function on (a,b) and $lim_{x \rightarrow a^{+}} f(x) = lim_{x \rightarrow b^{-}} f(x) = \infty.$ prove that f has a minimum on all of (a,b)

## The Attempt at a Solution

I have not tried an actual attempt yet. The only think I can think of doing is making two sequences that approach a common point on the domain of f. One sequence starting at a, and the other starting at b. Then show that the range of these sequences is decreasing and tends to the same value. This seems a bit too complicated to me for such a problem.

I am interested in where to start. Logically, it makes sense to me that there should be a minimum. I just don't know how to explain it using math.

Thanks.

A good way to think about these problems is that you have a couple different regions: if x is close to a, or close to b, then you know f(x) is really big. If f is not close to either of those, all you really know is that f is continuous. So you need to use something (probably a theorem) involving the continuity of f to make a statement about f in this region that I have vaguely described as "not close to a or b". You of course should make that description mathematically more precise first!

If you start at a and travel towards b, you will have that ##f(x) < lim_{x \rightarrow a^+}f(x) = \infty##. Yet by the time you get back to ##b^-## you are back at ## \infty ##. How did that happen?

If you are not sure where the continuity fits in, try constructing a function which goes to ## \infty## at a and b, but does not have a minimum on (a,b).